Autonomy is one of the major concerns during the planning of a space mission, whether its objective is scientiﬁc (interplanetary exploration, observations, etc.) or commercial (service in orbit). For space rendezvous, this autonomy depends on the on-board capacity of controlling the relative movement between two spacecraft. In the context of satellite servicing (trou- bleshooting, propellant refueling, orbit correction, end-of-life deorbit, etc.), the feasibility of such missions is also strongly linked to the ability of the guidance andcontrolalgorithms to account for all operational constraints (for example, thruster saturation or restrictions on the relative positioning between the vehicles) while maximizing the life of the vehicle (minimizing propellant consumption). The literature shows that this problem has been intensively studied since the early 2000s. However, the proposed algorithms are not entirely satisfactory. Some approaches, for example, degrade the constraints in order to be able to base thecontrol algorithm on an eﬃcient optimization problem. Other methods accounting forthe whole set of constraints of the problem are too cumbersome to be embedded on real computers existing in the spaceships.
Recent space missions rely more and more on the cooperation between different spacecraft in order to achieve a desired objective. Among thespacecraft proximity operations, the orbital rendezvous is a classical example that has generated a large amount of studies since the beginning of the space exploration. However, the motivations and objectives forthe proximity operations have considerably changed. The need for higher autonomy, better security and lower costs prompts forthe development of new guidance andcontrolalgorithms. The presence of different types of constraints and physical limitations also contributes to the increased complexity of the problem. In this challenging context, this dissertation represents a contribution to the development of new spacecraft guidance andcontrolalgorithms.
In the literature, fuel-optimal spacecraft flying-formation problems and model predictive controllers related to space applications are traditionally formulated as Linear Programs (LP) (see [10, 16, 15]) thanks to discretization procedures. In the works of Deaconu and Louembet (see [5, 4]), a different approach for modeling the space constraints of therendezvous problems was proposed. It consists in characterizing the periodic relative trajectories that are enclosed by polytopes defined in the local frame of the leader satellite by polynomials non-negativity constraints and employing a result demonstrated by Nesterov  to convert these polynomial constraints into linear matrix inequalities.
Forspacecraft proximity operations (spacecraftrendezvous, station keeping, collision avoidance), the relative dynamics are often linearized for both propagation or control purposes. More specifically, when the magnitude of the relative motion of thespacecraft is small compared to its distance to the Earth, one linearizes the equations of motion, which implies solving simpler linear di fferential equations. However, no closed form solution is available for these equations in most cases. Exceptionally, for instance, Tschauner-Hempel equations for linearized Keplerian relative motion  admit an analytical solution forthe transition matrix . Forspacecraft station keeping on geostationary Earth orbits (GEO), disturbing effects must be handled. Some models like the CNES Orange model  describe the orbital perturbations as the effect of the true geopotential, the lunisolar attraction andthe Sun radiation pressure. A transition matrix is not available in this setting, except forthe case when considering only the oblateness of the Earth in some cases, for which some analytical methods were developed forthe description of the relative motion [4, 5]. However, those methods are not applicable for geostationary spacecraft due to the numerical issues coming from the zero inclination of the reference orbit.
Impulsive zone model predictive controlforrendezvous hovering phases
Christophe Louembet 1 , Alejandro H. González 2 and Paulo R. Arantes Gilz 1
Abstract— In this manuscript, an impulsive zone MPC for- mulation is proposed to tackle the problem of thespacecraftrendezvouscontrol. Thecontrol objective is to maintain the follower spacecraft in a given subspace with respect to a leader vehicle by stabilizing the set of periodic relative orbits included in a given hovering zone. The idea is to incorporate this hov- ering zone as a target set into the MPC cost function, in order to permit a single MPC formulation and a receding horizon implementation. Thecontrol algorithm takes advantages of a relative motion parametrization for which the set of the equilibrium states represent the set of periodic orbits to prove the stability of the hovering zone and to enlarge significantly the domain of attraction. Several simulation results show that, in addition, performances in terms of convergence and fuel consumption are improved in comparison with previous works.
Future works should focus in investigating the robustness of the proposed controller from a theoretical point of view, providing, for example, an idea of the influence of the nonlinearities, disturbances and scenario parameters on the stability of the method.
An extension of this work could combine safety requirements such as collision avoidance, passive safety or visibility, with our proposed station-keeping algorithm. This is due to the fact that existing guidance algorithms [ 35 ] which handle these constraints have the same mathematical formalism as the constrained optimization presented in this work. This can be done by considering time-varying path constraints, but this would imply revisiting the stability analysis while accounting for such time-varying constraints.
Keywords: satellite control, time-varying systems, Lyapunov stability
The last couple of decades have seen an astonishing series of achievements in aerospace science and technology, such as the increased deployment of reusable launch vehicles and nano-satellites, that have marked the beginning of an exciting new space era. These developments have led to the definition of new mission scenarios that necessitate more efficient hardware components and robust algorithms which however should not increase the overall system com- plexity and cost. Such requirements influence directly one of the most critical components forthe precise operation of a spacecraft. This is the attitude control system (ACS) that ensures the active attitude stabilization and distur- bance rejection. The first immediate task of the ACS after launch is to detumble thespacecraft, i.e. drive all angular velocities to zero.
The Fringe Locking (FL) system resulting from a combination of an optical arrangement and an electronic feedback control shown in Chapter 1, is another good example of this kind of solution. A variant of this configuration using two servo stabilized LD’s working in di fferential mode resulted in a speckle insensitive proposal allowing to provide a large dynamic range of amplitude measurement [ 96 ]. In a recent study from the same research group [ 97 ], it is informed that frequency compensation of the wavelength-modulation in a pump current controlled LD shall lead to an optimized analog control loop and thus an increase of performance. As a side note, the commercial SM sensor from Philips mentioned in Chapter’s 1 introduction, operates the LD with a triangular current modulation to reveal the direction of target surface motion by comparison of the measured Doppler shift on the rising and falling slopes in the waveform [ 17 ]. A di fferent elaboration consist in acting on the focus of the system by means of a liquid lens (LL) as proposed in [ 98 ]. The idea is to maintain the SM phenomenon in moderate feedback regime with no hysteresis in a dynamic way thanks to the adaptive optical element. The setup was mentioned to also increase the dynamic range of the distance between the target and LD. Such a system reported a good performance over a distance range of 6.5 cm to 265 cm for a target vibration signal reconstitution.
Abdelkrim Achaibou ‡
LAAS du CNRS, Toulouse, FRANCE
The delegation to the flight crew of some tasks currently performed by air traffic controllers provides new perspectives to potentially increase air traffic control efficiency. More specifically, the task of establishing properly spaced landing sequences is very demanding in heavy traffic conditions forthe air traffic controllers in charge of the terminal maneuvering area. Automatic merging and station keeping operations could relieve air traffic controller of providing time consuming radar vectoring instructions. The objective of this communication is to provide technical insight into the airborne devices andalgorithms which may be used onboard aircraft to automatically achieve a specified distance or delay with respect to another aircraft at a specified meter fix. A nonlinear control law based on sliding mode control is proposed to controlthe lateral motion of the trailing aircraft. The design is followed by two illustrative examples which show the effectiveness of the proposed approach.
commands the local actuators (u L and u F ):
u L = K L (s) · y L , u F = K F (s) · y F
Decentralized control is a vast ﬁeld of ongoing research which has also reached the area of formation ﬂying control, cf. . The problem of ﬁnding an optimal decentralized controller in the sense of a H 2 criterion usually involves solving a linear matrix inequality (LMI) problem subject to bilinear matrix inequality (BMI) constraints, which is a non-convex problem and thus generally leads to local minima, cf. . Although algorithmsforthe solution of this type of problem exist, the handling of BMI problems remains intricate. Under some conditions concerning the interconnection structure of the system andthe controller, the problem may become convex, cf. . Another possibility is to directly optimize the controller parameters, as described in Ref. , an approach which does not, alas, remedy the problem of local minima.
This section briefly outlines the sequence of the flight software implemented on the onboard computers forthe two UAVs. When the computer is turned on, a RAM-disk is allotted forthe storage space of the flight data. The flight software starts next and performs a series of initializations. One of the important task in the initialization steps is the declaration of a software timer interrupt. Then the flight software follows the sequence outlined in Figure A- 11. The program enters the main loop in which the flight software waits forthe GPS data update that occurs at 5 Hz. If the GPS data is updated, a few functions are executed such as flight path generation, transceiver and copilot inputs handling, data storage, etc. On the other hand, after the definition of the timer interrupt, at every 0.025 seconds (40Hz) the timer interrupt defined in the initialization step begins to occur and a few time-critical tasks are performed. These include the reading the analog flight sensors, the performing the estimation andthe controller algorithms, andthe sending thecontrol inputs to the servo motors. The program is terminated by the copilot input which is picked in the main loop of the software, andthe flight data is copied from the RAM-disk to the DiskOnChip of the CPU module.
To overcome these drawbacks, an event-triggered model predictive controller is considered in this work for station- keeping. Event-based control is a control methodology where the commands are asynchronously computed, re- ducing the communication needs between the sensors, the on-board computer andthe actuators in thecontrol loop (see Astr¨ om (2008) forthe basics). This methodology can be combined with feedback policies, see Wu et al. (2014) and references therein, and model-predictive schemes, see Pawlowski et al. (2015). In the context of spacecraft oper- ations, event based controllers are recently attracting the attention of the attitude control community, see Wu et al. (2018) and Zhang et al. (2018), whereas some initial work forrendezvous hovering phases can be found in Louembet and Arantes Gilz (2018).
AND TECHNOLOGICAL CONSTRAINTS The relative motion consists in the motion of a chaser spacecraft equipped with thrusters with respect to the moving local frame attached to a passive target spacecraft. Under Keplerian assumptions, the relative motion between two spacecraft in the Earth gravitational field has been ex- pressed by means of linearized impulsive differential equa-
The simulator here discussed is a modified version of the Matlab
developed by Mounir Kara-Zaitri during his PhD thesis at the LAAS-CNRS. The implemented modifications were performed in order to obtain a better-suited tool for simulating and de- veloping controlalgorithmsforthe orbital spacecraftrendezvous without using the Matlab
knowledge of how some subsystem parameters affect their own requirements and other subsystems specifications. Due to the importance of an integrated control/subsys- tems design methodology, many attempts have been made, mainly in structural control literature, since the publi- cation of the first integrated design methodologies such as those in Onoda and Haftka (1987), Gilbert (1988) or Messac and Malek (1992). These methods were based on iterative methodologies with optimization algorithms. Lately, other methods have been proposed such as those solved by LMI algorithms or with LQG methods like in Hiramoto et al. (2009) and Cimellaro et al. (2008) respec- tively. However, these approaches give conservative results and their applicability is restricted by problem dimension. Recently, a counterpart technique currently under develop- ment in ONERA Toulouse Research Center allows a more general approach (Alazard et al., 2013). Actually, this method is based on structured H ∞ synthesis algorithms
scenario in practice (sec. 4).
The simulator here discussed is a modified version of the one developed by Mounir Kara-Zaitri during his PhD thesis at the LAAS-CNRS. The implemented modifications were performed in order to obtain a dedicated tool for simulating and developing controlalgorithmsforthe orbital spacecraftrendezvous in the case where the leader spacecraft is passive andthecontrol applied on follower spacecraft is originally computed on the leader LVLH frame.
Formation ﬂying of multiple spacecraft is an enabling technology for many future space science missions. These future missions will, for example, use the highly coor- dinated, distributed array of vehicles for earth mapping interferometers and synthetic aperture radar. This thesis presents coordination andcontrolalgorithms designed for a ﬂeet of spacecraft. These algorithms are embedded in a hierarchical ﬂeet archi- tecture that includes a high-level coordinator forthe ﬂeet maneuvers used to form, re-size, or re-target the formation conﬁguration and low-level controllers to generate and implement the individual control inputs for each vehicle. The trajectory andcontrol problems are posed as linear programming (LP) optimizations to solve forthe minimum fuel maneuvers. The combined result of the high-level coordination and low-level controllers is a very ﬂexible optimization framework that can be used oﬀ-line to analyze aspects of a mission design and in real-time as part of an on-board autonomous formation ﬂying control system. This thesis also investigates several crit- ical issues associated with the implementation of this formation ﬂying approach. In particular, modiﬁcations to the LP algorithms are presented to: include robustness to sensor noise, include actuator constraints, ensure that the optimization solutions are always feasible, and reduce the LP solution times. Furthermore, the dynamics forthecontrol problem are analyzed in terms of two key issues: 1) what dynamics model should be used to specify the desired state to maintain a passive aperture; and 2) what dynamics model should be used in the LP to represent the motion about this state. Several linearized models of the relative dynamics are considered in this analysis, including Hill’s equations for circular orbits, modiﬁed linear dynamics that partially account forthe J 2 eﬀects, and Lawden’s equations for eccentric orbits. The complete formation ﬂying control approach is successfully demonstrated using a nonlinear sim- ulation environment that includes realistic measurement noises, disturbances, and actuator nonlinearities.
Decomposition has been recognized as an important leverage to manage architectural complexity of the systems to be diagnosed. Most of the approaches, however, focus on hierarchical decomposition , while decentralization has been explored less frequently. The majority of decentralized diagnosis methods concern discrete event systems [5, 6, 7]. In , the purpose of the method is to provide efficient online diagnosis to detect and isolate faults in large discrete event systems.  uses a decentralized approach to deal with the size of the model and to get a tractable representation of diagnosis. Along the same idea,  proposes a hierarchical framework that exploits different local decisions.
V. TESTS ON KNOWN MATERIALS
The 3 Omega method, as implemented by AST-DE, was tested on several materials with known properties for verification. First, a sample of Rohacell which was tested by the Physikalische Technische Bundesanstalt in Germany, with measured conductivity of 32 mW/m∙K ± 10% and diffusivity of 0.26 mm 2 /s ± 15%. The results of measurements performed at AST are shown in Fig. 9 resulting in measured conductivity of 29.9 mW/m∙K ± 4% and diffusivity of 0.22 mm 2 /s ± 17%. Increasing the heating power applied to the sample in the measurement process by a factor of 3 resulted in measurement values consistent within a few percent. The high probability of consistency with a constant (>0.05) indicates an over-estimate of the uncertainty in the measurements, such that true uncertainty must be lower. In Fig. 9 green lines show the nominal values forthe sample.
With the requirement of a reliable and safe-by-construction controller, NMPC prob- lems are solved by the help of validated simulation methods. They are mainly based on Taylor series [13, 14] or on Runge-Kutta methods [15, 16]. The latter is efficient in short simulation with interval initial values and parameters. Moreover, it is also embed- ded into the constraint satisfaction problems framework  offering new capabilities that are the requirements forthe synthesis of robust NMPC methods.